Presentation: Random walk models in biology E.A.Codling et al. Journal of The Royal Society Interface R EVIEW March 2008 Random walk models in biology

Presentation: Random walk models in biology E.A.Codling et al.

Journal of The Royal Society Interface

REVIEWMarch 2008

Random walk models in biology

Edward A. Codling1,*, Michael J. Plank2 and Simon Benhamou3

1Department of Mathematics, University of Essex, Colchester CO4 3SQ, UK2Department of Mathematics and Statistics, University of Canterbury, Christchurch 8140,

New Zealand3Behavioural Ecology Group, CEFE, CNRS, Montpellier 34293, France

Presented by Oleg Kolgushev

Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/01/30 - 2011/01/30


• Introduction to Random walk theory• Fundamentals of Random walks

– Simple (isotropic) Random Walks (SRWs)– Biased Random Walks (BRWs), waiting times, higher dimensions– Spatially dependent movement probabilities, Fokker–Planck equation– General diffusive properties and model limitations– Random walk with barriers– Correlated Random Walks (CRWs) and the telegraph equation

• Random walks as models of biological organism movement– Mean squared displacement of CRWs– Mean dispersal distance of unbiased CRWs– Tortuosity of CRWs– Bias in observed paths– Reinforced random walks– Biological orientation mechanisms

• Conclusion and future work

Contents



• RWs are traced back to Brownian motion and classical workson probability theory

• Physicists extended RWs into many important fields:random processes, random noise, spectral analysis, stochastic equations

• The first simple models of movement using random isotropicwalks are uncorrelated and unbiased (SRWs is a basis of most diffusive processes)

• Correlated random walks (CRWs) involve a “persistence” between successive step orientations (local bias)

• Paths with global bias in the preferred direction (target) are termed Biased Random Walks (BRWs)

• CRWs and BRWs produce BCRWs

• RW theory is applied in 2 main biological contexts: the movement and dispersal of organisms, and chemotaxis models of cell signaling and movement.

Introduction


http://upload.wikimedia.org/wikipedia/commons/8/8c/Random_walk_in2D.png


• A walker moving on an infinite one-dimensional (x) uniform lattice. Motion is random. x0 = 0, ∆x = δ, ∆t = τ

• The probability that a walker isat mδ to the right of the origin after n time steps (even) (2.1)

• Taking the limit δ, τ -> 0 such that δ2/τ = 2D gives the Probability DensityFunction (PDF) of walker location(diffusion equation) (2.2)

• The mean location and the meansquared displacement (MSD) defined by (2.3)

Simple Random Walks



• A walker moving on an infinite one-dimensional (x) uniform lattice with probabilities moving to right r, left – l, and not moving (1-l-r), x0 = 0, ∆x = δ, ∆t = τ

• Taking the limit δ, τ -> 0 and rewriting (2.4) as Taylor series about (x,t) gives partial differential equation (PDE) (2.5)where ϵ=r-l; κ=l+r;

• Let exists condition (2.6)

• Under these limits higher order termsin (2.5) tend to zero, giving (2.7) (drift-diffusion equation)

Solution of (2.7) with initial conditionp(x,0)= δD(x) (Dirac delta function) is

Biased Random Walks



• Substituting (2.8) into (2.3) we can get MSD

• In contract with SRW the MSD is proportional to t2 so the movement propagates as wave and more appropriate measure is the dispersal about the origin

• A similar definitions can be extended into N-dimensional lattice giving standard drift-diffusion equation(where u is the average drift velocity) with solution (2.12)

The mean location and MSD are calculated in similar way

Biased Random Walks



Biased Random Walks



Spatially Dependant Movement



General diffusive properties and model limitations



Random walk with barriers



Correlated Random Walks and the telegraph equation



Mean squared displacement of CRWs



Mean dispersal distance of unbiased CRWs



Tortuosity of CRWs



Bias in observed paths



Reinforced random walks



Biological orientation mechanisms



Conclusion



• Introduction to Random walk theory• Fundamentals of Random walks

– Simple isotropic Random Walk (SRW)

References



• Bias: preference for moving in a particular direction.• Isotropic: uniform in all directions.• Sinuosity: a measure of the tortuosity of a random walk.• Tortuosity: the amount of turning associated with a path.• Central Limit Theorem: mean of a sufficiently large number of

independent random variables, each with finite mean and variance, will be approximately normally distributed

• Correlated random walk (CRW): random walk with persistence• Taxis: directional response to a stimulus (cf. kinesis). Examples

include chemotaxis, phototaxis, gyrotaxis.

Glossary


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Presentation: Random walk models in biology E.A.Codling et al. Journal of The Royal Society Interface R EVIEW March 2008 Random walk models in biology